198 ANALYTICAL MECHANICS
fore the total change in the potential energy of the system equals the sum of the work done, during the rearrangement,, on all the particles of the system; i.e.,
where the summation covers all the particles of the system. Therefore substituting the expression for FT, which was obtained by applying the law of action and reaction to the motion of particles, we obtain
= —2ml vdv
where TQ and T are, respectively, the values of the total kinetic energy of the. system in the initial and in the final configurations. Rearranging the terms of the last equation we get
U+ T= UQ + TQ= const.
which is the principle of the conservation of dynamical energy. Therefore the principle of the conservation of dynamical energy and the law of force are not independent of each other but form two different aspects of the same universal principle.
ILLUSTRATIVE EXAMPLE.
Taking into account the variation of the gravitational attraction with the distance of a body from the center of the earth, find the potential energy of a body with respect to the surface of the earth.
Outside the earth the weight of a body varies inversely as the square of its distance from the center of the earth. Therefore denoting this variable weight by F we have
Tf k Jb = — i